How do you write the given equation \[{x^2} - {y^2} = 1\] into polar form?

Hint: Here we will simply substitute the value of \[x = r\cos \theta ,y = r\sin \theta \] in the equation and solve it. Then after solving it we will write it in the simplified form which will be the polar form of the given equation.

Complete Step by Step Solution:
Given equation is \[{x^2} - {y^2} = 1\].
So, to convert the given equation into the polar form we will substitute \[x = r\cos \theta ,y = r\sin \theta \]. Therefore the equation becomes
\[ \Rightarrow {\left( {r\cos \theta } \right)^2} - {\left( {r\sin \theta } \right)^2} = 1\]
Now we will simply open the brackets and square the terms in the bracket. Therefore, we get
\[ \Rightarrow {r^2}{\cos ^2}\theta - {r^2}{\sin ^2}\theta = 1\]
Now we will take \[{r^2}\] common from the terms of the equation, we get
\[ \Rightarrow {r^2}\left( {{{\cos }^2}\theta - {{\sin }^2}\theta } \right) = 1\]
We know that \[{\cos ^2}\theta - {\sin ^2}\theta = \cos 2\theta \]. Therefore, we will put this value in the above equation. Therefore, the equation becomes
\[ \Rightarrow {r^2}\left( {\cos 2\theta } \right) = 1\]
Now we will take the term \[\cos 2\theta \] to the other side of the equation, we get
\[ \Rightarrow {r^2} = \dfrac{1}{{\cos 2\theta }}\]
We know that the reciprocal of the cos trigonometric function is equal to sec trigonometric function. Therefore the equation becomes
\[ \Rightarrow {r^2} = \sec 2\theta \]

Hence the given equation \[{\left( {x - 3} \right)^2} + {y^2} = 9\] into polar form can be written as \[{r^2} = \sec 2\theta \].

Note:
Here we have to note that the ratio of the \[\sin \theta \] and \[\cos \theta \] is equal to the \[\tan \theta \]. Also the ratio of \[\cos \theta \] and \[\sin \theta \] is equal to \[\cot \theta \] and the reciprocal of the cos function is equal to sec function and reciprocal of the sin function is equal to the cosec function.
\[\tan A = \dfrac{{\sin A}}{{\cos A}},\sec A = \dfrac{1}{{\cos A}},\cos ecA = \dfrac{1}{{\sin A}}\]
We don’t have to confuse the polar coordinate system with the normal rectangular coordinate system. Polar coordinate system is the system in which the coordinates of a point is represented by the distance of that point from a reference point and by the angle from the reference plane.